Bobenko a at al (eds ) discrete differential geometry

Oberwolfach Seminars Volume 38 Discrete Differential Geometry Alexander I Bobenko Peter Schröder John M Sullivan Günter M Ziegler Editors Birkhäuser Basel · Boston · Berlin Alexander I Bobenko Institut für Mathematik, MA 8-3 Technische Universität Berlin Strasse des 17 Juni 136 10623 Berlin, Germany e-mail: bobenko@math.tu-berlin.de John M Sullivan Institut für Mathematik, MA 3-2 Technische Universität Berlin Strasse des 17 Juni 136 10623 Berlin, Germany e-mail: sullivan@math.tu-berlin.de Peter Schröder Department of Computer Science Caltech, MS 256-80 1200 E California Blvd Pasadena, CA 91125, USA e-mail: ps@cs.caltech.edu Günter M Ziegler Institut für Mathematik, MA 6-2 Technische Universität Berlin Strasse des 17 Juni 136 10623 Berlin, Germany e-mail: ziegler@math.tu-berlin.de 2000 Mathematics Subject Classification: 53-02 (primary); 52-02, 53-06, 52-06 Library of Congress Control Number: 2007941037 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at ISBN 978-3-7643-8620-7 Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks For any kind of use permission of the copyright owner must be obtained © 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media 3ULQWHGRQDFLGIUHHSDSHUSURGXFHGIURPFKORULQHIUHHSXOS7&)’ Printed in Germany ISBN 978-3-7643-8620-7 e-ISBN 978-3-7643-8621-4 987654321 www.birkhauser.ch Preface Discrete differential geometry (DDG) is a new and active mathematical terrain where differential geometry (providing the classical theory of smooth manifolds) interacts with discrete geometry (concerned with polytopes, simplicial complexes, etc.), using tools and ideas from all parts of mathematics DDG aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields such as computer graphics Discrete differential geometry initially arose from the observation that when a notion from smooth geometry (such as that of a minimal surface) is discretized “properly”, the discrete objects are not merely approximations of the smooth ones, but have special properties of their own, which make them form a coherent entity by themselves One might suggest many different reasonable discretizations with the same smooth limit Among these, which one is the best? From the theoretical point of view, the best discretization is the one which preserves the fundamental properties of the smooth theory Often such a discretization clarifies the structures of the smooth theory and possesses important connections to other fields of mathematics, for instance to projective geometry, integrable systems, algebraic geometry, or complex analysis The discrete theory is in a sense the more fundamental one: the smooth theory can always be recovered as a limit, while it is a nontrivial problem to find which discretization has the desired properties The problems considered in discrete differential geometry are numerous and include in particular: discrete notions of curvature, special classes of discrete surfaces (such as those with constant curvature), cubical complexes (including quad-meshes), discrete analogs of special parametrization of surfaces (such as conformal and curvature-line parametrizations), the existence and rigidity of polyhedral surfaces (for example, of a given combinatorial type), discrete analogs of various functionals (such as bending energy), and approximation theory Since computers work with discrete representations of data, it is no surprise that many of the applications of DDG are found within computer science, particularly in the areas of computational geometry, graphics and geometry processing Despite much effort by various individuals with exceptional scientific breadth, large gaps remain between the various mathematical subcommunities working in discrete differential geometry The scientific opportunities and potential applications here are very substantial The goal of the Oberwolfach Seminar “Discrete Differential Geometry” held in May–June 2004 was to bring together mathematicians from various subcommunities vi Preface working in different aspects of DDG to give lecture courses addressed to a general mathematical audience The seminar was primarily addressed to students and postdocs, but some more senior specialists working in the field also participated There were four main lecture courses given by the editors of this volume, corresponding to the four parts of this book: I: II: III: IV: Discretization of Surfaces: Special Classes and Parametrizations, Curvatures of Discrete Curves and Surfaces, Geometric Realizations of Combinatorial Surfaces, Geometry Processing and Modeling with Discrete Differential Geometry These courses were complemented by related lectures by other participants The topics were chosen to cover (as much as possible) the whole spectrum of DDG—from differential geometry and discrete geometry to applications in geometry processing Part I of this book focuses on special discretizations of surfaces, including those related to integrable systems Bobenko’s “Surfaces from Circles” discusses several ways to discretize surfaces in terms of circles and spheres, in particular a Măobius-invariant discretization of Willmore energy and S-isothermic discrete minimal surfaces The latter are explored in more detail, with many examples, in Băuckings article Pinkall constructs discrete surfaces of constant negative curvature, documenting an interactive computer tool that works in real time The final three articles focus on connections between quadsurfaces and integrable systems: Schief, Bobenko and Hoffmann consider the rigidity of quad-surfaces; Hoffmann constructs discrete versions of the smoke-ring flow and Hashimoto surfaces; and Suris considers discrete holomorphic and harmonic functions on quadgraphs Part II considers discretizations of the usual notions of curvature for curves and surfaces in space Sullivan’s “Curves of Finite Total Curvature” gives a unified treatment of curvatures for smooth and polygonal curves in the framework of such FTC curves The article by Denne and Sullivan considers isotopy and convergence results for FTC graphs, with applications to geometric knot theory Sullivan’s “Curvatures of Smooth and Discrete Surfaces” introduces different discretizations of Gauss and mean curvature for polyhedral surfaces from the point of view of preserving integral curvature relations Part III considers the question of realizability: which polyhedral surfaces can be embedded in space with flat faces Ziegler’s “Polyhedral Surfaces of High Genus” describes constructions of triangulated surfaces with n vertices having genus O.n2 / (not known to be realizable) or genus O.n log n/ (realizable) Timmreck gives some new criteria which could be used to show surfaces are not realizable Lutz discusses automated methods to enumerate triangulated surfaces and to search for realizations Bokowski discusses heuristic methods for finding realizations, which he has used by hand Part IV focuses on applications of discrete differential geometry Schrăoders What Can We Measure?” gives an overview of intrinsic volumes, Steiner’s formula and Hadwiger’s theorem Wardetzky shows that normal convergence of polyhedral surfaces to a smooth limit suffices to get convergence of area and of mean curvature as defined by the Preface vii cotangent formula Desbrun, Kanso and Tong discuss the use of a discrete exterior calculus for computational modeling Grinspun considers a discrete model, based on bending energy, for thin shells We wish to express our gratitude to the Mathematisches Forschungsinstitut Oberwolfach for providing the perfect setting for the seminar in 2004 Our work in discrete differential geometry has also been supported by the Deutsche Forschungsgemeinschaft (DFG), as well as other funding agencies In particular, the DFG Research Unit “Polyhedral Surfaces”, based at the Technische Universităat Berlin since 2005, has provided direct support to the three of us (Bobenko, Sullivan, Ziegler) based in Berlin, as well as to Băucking and Lutz Further authors including Hoffmann, Schief, Suris and Timmreck have worked closely with this Research Unit; the DFG also supported Hoffmann through a Heisenberg Fellowship The DFG Research Center M ATHEON in Berlin, through its Application Area F “Visualization”, has supported work on the applications of discrete differential geometry Support from M ATHEON went to authors Băucking and Wardetzky as well as to the three of us in Berlin The National Science Foundation supported the work of Grinspun and Schrăoder, as detailed in the acknowledgments in their articles Our hope is that this book will stimulate the interest of other mathematicians to work in the field of discrete differential geometry, which we nd so fascinating Alexander I Bobenko Peter Schrăoder John M Sullivan Găunter M Ziegler Berlin, September 2007 Contents Preface v Part I: Discretization of Surfaces: Special Classes and Parametrizations Surfaces from Circles by Alexander I Bobenko Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples by Ulrike Băucking 37 Designing Cylinders with Constant Negative Curvature by Ulrich Pinkall 57 On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces by Wolfgang K Schief, Alexander I Bobenko and Tim Hoffmann 67 Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow by Tim Hoffmann 95 The Discrete Green’s Function by Yuri B Suris 117 Part II: Curvatures of Discrete Curves and Surfaces 135 Curves of Finite Total Curvature by John M Sullivan 137 Convergence and Isotopy Type for Graphs of Finite Total Curvature by Elizabeth Denne and John M Sullivan 163 Curvatures of Smooth and Discrete Surfaces by John M Sullivan 175 Part III: Geometric Realizations of Combinatorial Surfaces 189 Polyhedral Surfaces of High Genus by Găunter M Ziegler 191 Necessary Conditions for Geometric Realizability of Simplicial Complexes by Dagmar Timmreck 215 x Contents Enumeration and Random Realization of Triangulated Surfaces by Frank H Lutz 235 On Heuristic Methods for Finding Realizations of Surfaces by Jăurgen Bokowski 255 Part IV: Geometry Processing and Modeling with Discrete Differential Geometry 261 What Can We Measure? by Peter Schrăoder 263 Convergence of the Cotangent Formula: An Overview by Max Wardetzky 275 Discrete Differential Forms for Computational Modeling by Mathieu Desbrun, Eva Kanso and Yiying Tong 287 A Discrete Model of Thin Shells by Eitan Grinspun 325 Index 339 Part I Discretization of Surfaces: Special Classes and Parametrizations 326 Eitan Grinspun F IGURE The local coordinate frame in a small neighborhood of a thin shell: two axes span the middle surface, and the normal shell director spans the thickness (e.g., hats, cans, carton boxes, pans, car bodies, detergent bottles) are thin shells and cannot be modeled using plate formulations Thin shells are remarkably difficult to simulate Because of their degeneracy in one dimension, shells not admit to straightforward tessellation and treatment as threedimensional solids; indeed, the numerics of such approaches become catastrophically illconditioned, foiling numerical convergence and accuracy Robust finite-element methods for thin-shell equations continue to be an active and challenging research area This chapter, which is based on [18], develops a simple model for thin shells with applications to computer animation Our discrete model of shells captures the same characteristic behaviors as more complex models, with a surprisingly simple implementation The model described here was previously demonstrated to be useful in computer animation; its ability to capture the qualitative behavior of thin shells was evaluated in various experiments, including comparisons to real-world shells (see Figure 3) Kinematics Since it is thin, the geometry of the shell is well described by its middle surface (see Figure 2) At any point on the middle surface the local tangent plane and surface normal induce a coordinate frame in which to describe “motion along the surface” and “motion along thickness” In the discrete setting, the topology of the middle surface is represented by the combinatorics of an oriented 2-manifold simplicial complex, M D fv; e; f g, where v D fv1 ; v2 ; : : :g, e D fe1 ; e2 ; : : :g, f D ff1 ; f2 ; : : :g are sets of vertices, edges and faces, respectively The geometry of the middle surface is given by the discrete configuration map, C W v 7! R3 , which assigns to every vertex, vi , a position, C.vi /, in the ambient space Together M and C correspond to the usual notion of a triangle mesh in R3 ; in our exposition we assume fixed combinatorics M , and discuss a temporally evolving configuration, Ct where the subscript refers to a specific instant in time A Discrete Model of Thin Shells 327 Restricting our attention to elastic (“memory-less”) materials, the physics can be understood in terms of the undeformed configuration (the stress-free shape) and the deformed configuration (the stressed shape at the current instant in time), C0 and C1 , respectively The elastic response of a material depends on the change in shape of the elastic body, and on the constitutive laws that describe the restoring forces associated to this change in shape The former is a purely geometric quantity What is the change in shape between C0 and C1 ? Since rigid motions (translations and rotations) not affect shape, the answer must be invariant under composition of C0 (likewise C1 ) with any rigid-body transformation A simple theorem is that any reasonable measure of change in shape, or generalized strain, may be written as a function of only the edge lengths and dihedral angles of C0 and C1 The proof lies in showing that the configuration can be completely recovered from the edge lengths and dihedral angles, up to an unknown (but here inconsequential) rigid-body transformation We will also expect our measure of strain to be zero when shape has not changed, and nonzero whenever shape has changed In particular, strain should “see” any local change in shape Among the simplest forms of generalized strain that satisfy these requirements are two expressions that are evaluated at a specific edge ei Comparing C0 to C1 , let s e ei / be the difference in length of edge ei , and let s  ei / be the difference in dihedral angle at ei While these are perhaps the simplest possible measures of generalized strain, other more complex formulas can offer attendant advantages Recent research in discrete shell models has focused on functions evaluated over mesh faces which aggregate in one term the configuration of all the incident edge lengths and dihedral angles [13] Nevertheless, our goal here is to develop the simplest discrete model of thin shells capturing their qualitative elastic behavior Constitutive model Having defined the geometry of thin shells, we turn our attention to the governing physical equations The stored elastic energy of a thin shell is at the heart of the equations which govern its response to elastic deformations The stored energy, W ŒC0 ; C1 , should be a function of the local strain, integrated over the middle surface We choose the simplest expression for energy that is consistent with Hookean mechanics In 1676 Robert Hooke stated The power [sic] of any springy body is in the same proportion with the extension This statement was the birth of modern elasticity, which states that a first order approximation for the response of a material is a force proportional to strain, and, consequently (by the definition of work as force over distance), that the first approximation of stored energy is quadratic in strain We propose an energy with two kinds of terms, measuring stretching and bending modes, respectively: X X 2 kie s e ei / C kk s  ek / : WM ŒC0 ; C1  D ei 2M ek 2M 328 Eitan Grinspun This expression has several desirable properties First, it is positive whenever the shapes of C0 and C1 differ, and zero otherwise Second, evaluations over subsets of M satisfy the usual inclusion/exclusion principle: for A; B M , WM D WA CWB WA\B , which is consistent with continuum formulations in which energy is defined as an integral of energy density over the middle surface Third, because strain is invariant under rigidbody transformations of the undeformed and deformed configurations, Noether’s theorem guarantees that the resulting dynamics will conserve linear and angular momentum We offer the following interpretations for the above membrane and bending terms: Membrane Elastic surfaces resist stretching (local change in length) While some materials such as rubber sheets may undergo significant deformations in the stretching or shearing (membrane) modes, we focus on inextensible shells which are characterized by nearly isometric deformations, i.e., possibly significant deformations in bending but unnoticeable deformation in the membrane modes Most membrane models for triangle meshes satisfy this small-membrane-strain assumption with choice of suitably large membrane stiffness coefficient, kie Rewriting the membrane term in the following form permits an alternative interpretation: Ã2  e e e jei j W ei / D k jei j jeNi j/ D k jeNi j jeNi j where jei j is the length of edge i , quantities with a bar (such as eNi ) refer to the undeformed configuration C0 and remaining quantities are with respect to C1 ; note that we have dropped the subscript on kie indicating a uniform material stiffness over the domain of interest This is a unitless strain measurement, squared, and then integrated over the area of the local neighborhood, and multiplied by the material-dependent parameters Observe that under regular refinement of a triangle mesh, the local area indeed scales as jeNj j2 , which has units of area The units of the material parameters are energy per unit area, i.e., surface energy density In engineering models of shells, the material parameter is given as a volume energy density, and the energy is integrated over shell thickness yielding a surface energy density Efficient implementations of this formula precompute the quantities k e jeNi j2 , which depend only on the undeformed configuration Bending Consider the proposed discrete bending energy in relation to its continuous analogues Models in mechanics are typically based on tensors, and in particular shell models use the difference of the second fundamental forms in the deformed and undeformed configurations (pulling back the deformed tensor onto the undeformed configuration) These treatments derive tensorial expressions over smooth manifolds, and, as a final step, discretize the energy functional (or directly the forces) to carry out the numerics Our approach instead builds up a discrete energy functional by starting from a discrete shape operator The shape operator is the derivative of the Gauss map1 : geometrically, it measures the local curvature at a point on a smooth surface Our bending energy is an extrinsic measure of the difference between the shape operator evaluated on the deformed and This is the map from the surface to the unit sphere, mapping each surface point to its unit surface normal A Discrete Model of Thin Shells 329 F IGURE Real footage vs Simulation: top, a real hat is dropped on a table; bottom, our shell simulation captures the bending of the brim Notice that volumetric-elasticity, plate, or cloth simulations could not capture this behavior, while earlier work on shell simulation required significant implementation and expertise undeformed surfaces We express this difference as the squared difference of mean curvature: N D 4.H ı ' HN /2 ; (3.1) ŒTr.' S/ Tr.S/ N where S and S are the shape operators evaluated over the undeformed and deformed configurations, respectively; likewise, HN and H are the mean curvatures; ' S is the pullback of S onto N , and we use Tr.' S / D ' Tr.S/ D Tr.S / ı ' D H ı ' for a diffeomorphism ' This measure is extrinsic: it sees only changes in the embedding of the surface over the reference domain we find the continuous flexural energy in R R Integrating (3.1) N Next, we discretize this integral over the piecewise linear mesh HN /2 d A N 4.H ı ' that represents the shell We derive the discrete, integral mean-curvature squared operator as follows We first partition the undeformed surface into a disjoint union of diamond-shaped tiles, TN , associated to each mesh edge, e, as indicated on the side figure Following [23], one e he can use the barycenter of each triangle to define these regions—or, T alternatively, the circumcenters Over such a diamond, the mean R N ej N (for a proof see [10]) A curvature integral is TN HN d AN D Âj R N ej similar argument leads to: TN H ı ' HN /d AN D  Â/j N Using the notion of area-averaged value from [23], we deduce that N hN e , where hN e is the span of the undeformed tile, which is H ı ' HN /jTN D  Â/= one sixth of the sum of the heights of the two triangles sharing e N For a sufficiently fine, non-degenerate tessellation approximating a smooth surface, the average over a tile (converging pointwise to its continuous counterpart) squared is equal to the squared average, R N hN e leading to: TN H ı ' HN /2 d AN D  ÂN /2 jej= Comparing an argument presented in [23], we observe that there may be numerical advantages in using circumcenters instead of barycenters for the definition of the diamond 330 Eitan Grinspun tiles (except in triangles with obtuse angles) This affects the definition of hN e and of the lumped mass below Since we only need to compute these values for the undeformed shape, the implementation and performance only of initialization code would be affected Bobenko notes that when circumcenters are used, this formulation of discrete shells coincides (for flat undeformed configurations) with the derivation of the discrete Willmore energy based on circle packing [4] As we have just seen, we can express our discrete flexural energy as a summation over mesh edges, jeNk j ; (3.2) W  ek / D K  Âk ÂNk hN k θe where the term for edge ek is where Âk and ÂNk are corresponding complements of the dihedral angle of edge ek measured in the deformed and undeformed configuration, respectively, K  is the material bending stiffness, and hN k is a third of the average of the heights of the two triangles incident to the edge ek (see the appendix for another possible definition of hN k ) Note that the unit of K  is energy (not surface energy density) This formulation is consistent with the physical scaling laws of thin shells: if the (deformed and undeformed) geometry of a thin shell is uniformly scaled by along each axis, then surface area scales as as does the total membrane energy, however the total bending (curvature squared) energy is invariant under uniform scaling Following the reasoning for (3.1), we could have formed a second energy term taking the determinant instead of the trace of S This would lead to a difference of Gaussian curvatures, but this is always zero under isometric deformations (pure bending) This is not surprising, as Gaussian curvature is an intrinsic quantity, i.e., it is independent of the embedding of the two-dimensional surface into its ambient three-dimensional space In contrast, flexural energy measures extrinsic deformations e Dynamics A comprehensive treatment of the temporal evolution of a thin shell is beyond the scope of this chapter; we briefly summarize the basic components required to simulate the motion of thin shells Our dynamic system is governed by the ordinary differential equation of motion xR D M rW x/ where x is the vector of unknown DOFs (i.e., the vertices of the deformed geometry) and M is the mass matrix We use the conventional simplifying hypothesis that the mass distribution is lumped at vertices: the matrix M is then diagonal, and the mass assigned to a vertex is a third of the total area of the incident triangles, scaled by the area mass density Newmark time-stepping We adopt the Newmark scheme [24] for ODE integration, xi C1 D xi C ti xP i C ti2 1=2 xP i C1 D xP i C ti ˇ/Rxi C ˇ xR i C1 ; /Rxi C xR i C1 ; A Discrete Model of Thin Shells 331 where ti is the duration of the i th timestep, xP i and xR i are configuration velocity and acceleration at the beginning of the i th timestep, respectively, and ˇ and are adjustable parameters linked to the accuracy and stability of the time scheme The numerical advantages of the Newmark scheme are discussed in [30] Newmark is either an explicit (ˇ D 0) or implicit (ˇ > 0) integrator: we used ˇ D 1=4 for final production, and ˇ D to aid in debugging Newmark gives control over numerical damping via its second parameter We obtained the best results by minimizing numerical damping ( D 1=2) Dissipation We model dissipation due to flexural oscillations by introducing a damping PN force proportional to ÂP Â/r x  , where rx  is the gradient of the bending angle with respect to the mesh position For elastic deformations, ÂPN D 0; for plastoelastic materials, ÂPN is in general nonzero This approach may be viewed as a generalization of Rayleigh damping forces based on the strain rate tensor [2] Discussion This discrete flexural energy (3.2) generalizes established formulations for (flat) plates both continuous and discrete: (a) Ge and coworkers presented a geometric arR gument that the stored energy of a continuous inextensible plate has the form N cH H C cK KdA for material-specific coefficients cH and cK [12]; (b) Haumann used a discrete hinge energy [20], similarly P Baraff and Witkin used a discrete constraint-based energy [2], of the form WB x/ D eN Âe2 Our approach generalizes both (a) and (b), and produces convincing simulations beyond the regime of thin plate and cloth models (see Section 5) The proposed discrete model has three salient features: (a) the energy is invariant under rigid-body transformation of both the undeformed and the deformed shape: our system conserves linear and angular momenta; (b) the piecewise nature of our geometry description is fully captured by the purely intrinsic membrane terms, and the purely extrinsic bending term; most importantly, (c) it is simple to implement Results We exercised our implementation on various problems, including fixed beams, falling hats, and pinned paper Computation time, on a 2GHz Pentium CPU, ranged from 0.25s– 3.0s per frame; timings are based on a research implementation that relies on automatic differentiation techniques Beams We pinned to a wall one end of a V-beam, and released it under gravity Figure demonstrates the effect of varying flexural stiffness on oscillation amplitude and frequency The flexural energy coefficient has a high dynamic range; extreme values (from pure-membrane to near-rigid) remain numerically and physically well behaved Observe that increasing flexural stiffness augments structural rigidity Compare the behavior of beams: the non-flat cross section of the V-beam contributes to structural rigidity This difference is most pronounced in the operating regime of low flexural stiffness (but high membrane stiffness) Here the material does not inherently resist bending, but a V-shaped cross section effectively converts a bending deformation into a stretching deformation One can mimic this experiment by holding a simple paper strip by its end; repeat after folding a V-shaped cross section 332 Eitan Grinspun F IGURE Three pairs of flat and V-beams with increasing flexural stiffness K  (left to right) of 100, 1000, and 10000; the non-flat cross section of the V-beam contributes to structural rigidity Elastic hats We dropped both real and virtual hats and compared (see Figure 3): the deformation is qualitatively the same, during impact, compression, and rebound Adjusting the damping parameter, we capture or damp away the brim’s vibrations Adjusting the flexural stiffness, we can make a hat made of hard rubber or textile: a nearly rigid hat or a floppy hat) Plastoelasticity As discussed in the early work of Kergosien and coworkers, a compelling simulation of paper would require a mechanical shell model [22] Using our simple shell model, we can easily simulate a sheet of paper that is rolled, then creased, then pinned (see Figure 5) Here the physics require plastic as well as elastic deformations We begin with a flat surface, and gradually increase the undeformed angles, ÂNe Notice: modifying the undeformed configuration effects a plastic deformation The kinematics of changing ÂNe span only physically realizable bending, i.e., inextensible plastic deformations In contrast, directly modifying xN could introduce plastic deformations with unwanted membrane modes We introduced elastic effects by applying three pin constraints to the deformed configuration Observe the half-crease on the left side The (plastically deformed) left and (untouched) right halves have incompatible undeformed shapes In this situation, there does not exist a globally consistent undeformed (strain-free) state Recent extensions More recently, we demonstrated that simple, discrete models of thin shells can also produce striking examples of shattering glass (see Figure 6) [13], and paper origami (see Figure 7) [5] Implementation An attractive practical aspect of the proposed model is that it may be easily incorporated into working code of a standard cloth or thin-plate simulator such as those commonly used by the computer graphics community [2] One must replace the bending energy with (3.2) From an implementation point of view, this involves minimal work For example, consider that [2] already required all the computations relating to Âe These and other implementation details were outlined in [18] A Discrete Model of Thin Shells 333 F IGURE Modeling a curled, creased, and pinned sheet of paper: by altering dihedral angles of the reference configuration, we effect plastic deformation While the rendering is texture-mapped we kept flat-shaded triangles to show the underlying mesh structure F IGURE A measure of discrete strain is used to fracture a thin shell in this simulation of a shattering lightbulb Further reading A comprehensive survey of this expansive body of literature is far beyond the scope of this chapter; as a starting point see [1, 9] and references therein Here we highlight only a few results from the graphics and engineering literature 334 Eitan Grinspun F IGURE Virtual origami: user-guided simulated folding of a paper sheet produces a classical origami dog Recently, novel numerical treatments of shells, significantly more robust than earlier approaches, have been introduced in mechanics by Cirak et al [8] and in graphics by Green et al [15] and Grinspun et al [19] among others These continuum-based approaches use the Kirchhoff–Love constitutive equations, whose energy captures curvature effects in curved coordinate frames; consequently, they model a rich variety of materials In contrast, thin plate equations assume a flat undeformed configuration Thin plate models are commonly used for cloth and garment simulations and have seen successful numerical treatment in the computer graphics literature (see [21] and references therein) Thin plates have also been useful for variational geometric modeling [6, 16, 29] and intuitive direct manipulation of surfaces [25, 26] In graphics, researchers have used two kinds of approaches to modeling plates: finite elements and mass-spring networks In the latter resistance to bending is effected by springs connected to opposite corners of adjacent mesh faces Unfortunately, this simple approach does not carry over to curved undeformed configurations: the diagonal springs are insensitive to the sign of the dihedral angles between faces In this chapter we have developed a very simple discrete model of thin shells One price that must be paid for this simplicity is that, while we have taken care to ensure the correct scaling factors for each energy term, for an arbitrary triangle mesh we cannot guarantee the convergence of this model to its continuum equivalent In [17] we present experimental results comparing the convergence of the discrete shell and other discrete curvature operators In [3, 27, 11], we consider nearly inextensible thin shells Under the assumption of isometric surface deformations, we show that the smooth thin-plate bendingy energy is quadratic in surface positions, and the smooth thin-shell bending energy is cubic in surface positions To carry these obesrvations over to the discrete setting, we build on an axiomatic treatment of discrete Laplace operators (further developed in [28]), deriving a family of discrete isometric bending models for triangulated surfaces in 3-space The resulting family of thin-plate (resp shell) discrete bending models has simple linear (resp quadratic) energy gradients and constant (resp linear) energy Hessians The simplicity of the energy gradients and Hessians enables fast time-integration of cloth [3] and shell A Discrete Model of Thin Shells 335 dynamics [11] and near-interactive rates for Willmore smoothing of large meshes [27] In a simulation environment, methods that assume isometric deformations must be coupled with an efficient technique for ensuring material inextensability In [14] we present an efficient algorithm, based on constrained Lagrangian mechanics, for projecting a given surface deformation onto the manifold of isometric deformations Acknowledgments The work described here is the fruit of a collaboration with Mathieu Desbrun, Anil Hirani, and Peter Schrăoder [18] References to recent work on modeling thin shells refer to an active collaboration with Mikl´os Bergou, Akash Garg, Yotam Gingold, Rony Goldenthal, David Harmon, Saurabh Mathur, Jason Reisman, Adrian Secord, Max Wardetzky, Zoăe Wood, and Denis Zorin The author is indebted to Alexander Bobenko, Jerry Marsden, and Anastasios Vayonakis for insightful discussions Pierre Alliez, Ilja Friedel, Charles Han, Jeff Han, Harper Langston, Anthony Santoro and Steven Schkolne were pivotal in the production of the images shown here The author is grateful for the generous support of the National Science Foundation (MSPA Award No IIS–05–28402, CSR Award No CNS–06–14770, CAREER Award No CCF–06–43268) References [1] Douglas Arnold, Questions on shell theory, Workshop on Elastic Shells: Modeling, Analysis, and Computation, 2000 [2] David Baraff and Andrew Witkin, Large steps in cloth simulation, Proceedings of SIGGRAPH, 1998, pp 43–54 [3] Mikl´os Bergou, Max Wardetzky, David Harmon, Denis Zorin, and Eitan Grinspun, A quadratic bending model for inextensible surfaces, Eurographics Symposium on Geometry Processing, June 2006, pp 227–230 [4] Alexander I Bobenko, A conformal energy for simplicial surfaces, Combinatorial and Computational Geometry, vol 52, MSRI Publications, August 2005, pp 133–143 [5] R Burgoon, E Grinspun, and Z Wood, Discrete shells origami, Proceedings of CATA (Seattle, WA, USA), 2006, pp 180–187 [6] George Celniker and Dave Gossard, Deformable curve and surface finite elements for freeform shape design, Computer Graphics (Proceedings of SIGGRAPH 91) 25 (1991), no 4, 257–266 [7] Philippe Ciarlet, Mathematical Elasticity Vol III, Studies in Mathematics and its Applications, vol 29, North Holland, Amsterdam, May 2000 [8] F Cirak, M Ortiz, and P Schrăoder, Subdivision surfaces: A new paradigm for thin-shell niteelement analysis, Internat J Numer Methods Engrg 47 (2000), no 12, 2039–2072 [9] F Cirak, M.J Scott, E.K Antonsson, M Ortiz, and P Schrăoder, Integrated modeling, niteelement analysis, and engineering design for thin-shell structures using subdivision, Computer Aided Design 34 (2002), no 2, 137–148 [10] David Cohen-Steiner and Jean-Marie Morvan, Restricted delaunay triangulations and normal cycle, Proceedings of the 19th Annual Symposium on Computational Geometry, 2003, pp 312–321 336 Eitan Grinspun [11] Akash Garg, Eitan Grinspun, Max Wardetzky, and Denis Zorin, Cubic shells, ACM/Eurographics Symposium on Computer Animation, Eurographics Association, 2007, pp 91–98 [12] Z Ge, H.P Kruse, and J.E Marsden, The limits of hamiltonian structures in three-dimensional elasticity, shells, and rods, Journal of Nonlinear Science (1996), 19–57 [13] Yotam Gingold, Adrian Secord, Jefferson Y Han, Eitan Grinspun, and Denis Zorin, Poster: A discrete model for inelastic deformation of thin shells, ACM/Eurographics Symposium on Computer Animation ’04 (Grenoble, France), 2004 [14] Rony Goldenthal, David Harmon, Raanan Fattal, Michel Bercovier, and Eitan Grinspun, Efficient simulation of inextensible cloth, ACM Trans Graph 26 (2007), no 3, 49 [15] Seth Green, George Turkiyyah, and Duane Storti, Subdivision-based multilevel methods for large scale engineering simulation of thin shells, Proceedings of ACM Solid Modeling, 2002, pp 265–272 [16] G Greiner, Variational design and fairing of 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Symposium on Geometry Processing, Jul 2007, pp 33–37 [29] William Welch and Andrew Witkin, Variational surface modeling, Computer Graphics (Proceedings of SIGGRAPH 92) 26 (1992), no 2, 157–166 A Discrete Model of Thin Shells 337 [30] M West, C Kane, J.E Marsden, and M Ortiz, Variational integrators, the Newmark scheme, and dissipative systems, International Conference on Differential Equations 1999 (Berlin), World Scientific, 2000, pp 1009–1011 Eitan Grinspun Department of Computer Science Columbia University 500 W 120th Street New York, NY, 10027 USA e-mail: eitan@cs.columbia.edu Index Alexander trick, 166 Altshuler–Steinberg determinant, 244 asymptotic net (discrete), 72 Băacklund transform, 99, 102, 124 discrete, 105, 107 bending energy (discrete), 16, 158, 330 Bianchi permutability, 101, 107 Bianchi surface, 84 Bishop frame, see parallel frame bounded variation (BV), 140 Cauchy arm lemma, 151 Cauchy/Crofton formula, 149 Cauchy–Riemann equations (discrete), 118–124 chain complex, 297, 302 Chakerian’s packing theorem, 151 Chebyshev net (discrete), 62, 78 Christoffel’s theorem, 26 circle packing/pattern, 23, 39 circular net, 17 cochain, coboundary, 299 codifferential (discrete), 313 Cohen-Steiner–Morvan formula, 272 cohomology, 306 Combescure transform, 72 combinatorial equivalence (of triangulations), 240 complex curvature, 97 discrete, 103 cone metric, cone angle, 277 conformal surface (discrete), 31 conjugate net, 17 discrete, 72 non-degenerate, 73 connection (discrete), 184 consistency (3D), 18, 123 cotangent formula, 182, 278, 313 Cs´asz´ar’s torus, 244 curvature (of curve), 137, 175 curvature force, 145 curvature tensor (discrete), 272 Darboux frame, 176 deformation cochain, 217, 220 deformation, conformal, 80 deformation, isometric, 67–91 finite, 73, 80 infinitesimal, 68 second-order, 81 deformed m-cube, 204 deleted product, 216 deRham complex, 303 differential form, 287–321 Dirichlet problem, 28, 277 discrete manifold, 297 distortion, 140, 152 double (of graph), 121 dressing, 99 dual cell decomposition, 121 dual isothermic surface, 23 dual lattice, 118 dual mesh, 308 Dyck’s regular map, 258 elastic curve, 98 discrete, 104, 112 elastic energy, 327 electromagnetism, 320 enumeration of triangulations, 235, 238–243 by vertex-splitting, 238 lexicographic, 239, 243 mixed lexicographic, 243 strongly connected, 238 essential arc (of knot), 170 Euler characteristic, 181, 193, 237, 268, 306 exponential function (discrete), 127 exterior derivative, 288 discrete, 300 f -vector, 192, 237 face (of polytope), 184, 206, 296 F´ary/Milnor theorem, 146, 148 Fenchel’s theorem, 7, 144 finite-element method (FEM), 277 finite total curvature (FTC) curve, 137, 143, 163 graph, 164 flexural energy, see bending energy flip (on quad-graph), 126 fluid mechanics, 320 force balance, 179 Fr´echet distance, 138 Frenet frame, 177 FTC, see finite total curvature Galerkin scheme, 277 Gauss–Bonnet theorem, 178 Gauss curvature, 3, 5, 57, 177 combinatorial, 181 discrete, 180, 185 340 Gauss map, 177 of K-surface, 59, 89 of minimal surface, 25–29, 38 geometric knot theory, 163, 170 geometric realization, 193, 215, 236, 244, 255 heuristic methods, 255–260 random, 248 with small coordinates, 249 Gergonne’s minimal surface, 44 graph/1-skeleton (of surface), 192 Grassmannian, 146 Green’s function (discrete), 130 Hadwiger’s theorem, 268 harmonic form, 316 harmonic function (discrete), 118, 184 Hashimoto flow/surface, 95–113, 158, 179 discrete, 103, 110 Heffter’s surfaces, 199 Hodge decomposition, 316 Hodge star, 311, 312 holomorphic function (discrete), 118–127 holonomy, 184 homology (simplicial), 304 inscribed polygon, 138, 169 integrability, 99, 124 intersection cocycle, 216, 218 intersection condition, 192 intersection number, 218 intrinsic/mixed volumes, 182, 267 inversive distance, 32 isomonodromic deformation, 132 isothermic surface (discrete), 19 isotopy, 163–173 isotropic Heisenberg magnet (IHM), 98 discrete, 103 Jordan length, 139 K-surface, 57–66 discrete, 61, 78 Gauss map of, 59, 89 with cone point, 62 with planar strip, 64 knotted graph, 164 Koebe polyhedron, 25, 38 labeling (of quad-graph edges), 122 Laplace–Beltrami operator, 178, 280 Laplacian, 316 convergence, 281 discrete, 118, 278 Lelieuvre formula (discrete), 83 Index length (definition), 139 Levi-Civita connection, 185 linking number, 221 Liouville’s theorem, locally flat, 169 logarithm (discrete), 129 matroid, see oriented matroid Maxwell’s equations, 320 McMullen–Schulz–Wills surfaces, 202 mean curvature, 5, 281 convergence, 282 discrete, 181, 272, 282 squared difference of, 329 vector, 177 mean width, 265 measure, 140, 264 medial axis, 279 mesh (of polygon), 139 metacyclic group, 201 metric distortion tensor, 280 minimal surface (discrete), 23, 182, 276 construction, 38–43 examples, 43–54 fundamental piece, 40 S-isothermic, 23 with polygonal boundary, 52 Miquel’s theorem, 18 mirror complex, 204 Măobius strip, 232 Măobius torus, 197, 237 Măobius transformation, 4, 108, 110 Morse theory, 202 Moutard equation (discrete), 83 mutation (in pseudoline arrangement), 257 neighborly cubical polytope, 209 neighborly surface, 192, 215, 244 Ringel’s, 195 Neovius’s surface, 47 Newmark time-stepping, 330 nodal basis, 277 non-inscribable polyhedron, nonlinear Schrăodinger equation (NLSE), 95 discrete, 104 nonlinear -model, 91 normal convergence, 279, 284 normal graph, 279 Novik’s ansatz, 217 obstruction polytope, 230 oriented matroid, 256 Index parallel frame, 96, 102, 176 plantri, 235 plastoelasticity, 332 Poincar´e lemma, 302 projection of curve, 146 of polytope, 206 pseudoform, 293 pseudoline arrangement, 257 pseudomanifold, 241 quad-graph, 17, 120 quasicrystalline, 125 rhombic embedding, 121 quaternion, 9, 96 reach, 279 realization, see geometric realization reciprocal-parallel, 70 rectifiable curve, 139, 164 recycling (of coordinates), 248 ridge (of complex), 184 rigid-motion invariant, 264 Ringel’s construction, 195 rotation scheme, 196 rubber band, 257 S-isothermic surface, 22, 38 S-quad-graph, 22 Schewe’s non-realizable surfaces, 194, 215, 247, 255 Schlegel diagram, 209 Schoen’s I-6-surface, 51 Schur’s comparison theorem, 150 Schwarz lantern, 141, 276 Schwarz minimal surfaces CLP-surface, 46 D-surface, 47 H-surface, 50 P-surface, 29 Schwarz reflection, 40 shape operator, 177, 328 shortest distance map, 279 simplicial complex, 216, 296 simplicial embedding, see geometric realization simplicial map, 216 smoke-ring flow, see Hashimoto flow/surface Sobolev space (on polyhedra), 278 Steiner’s formula, 182, 269 stereographic projection, 30 stretchability (of pseudoline arrangement), 257 strictly preserved face, 206 strip (in quad-graph), 121 341 strongly connected, 238 surface combinatorial, 192 construction problem, 193 geometric, see surface, polyhedral of high genus, 191 polyhedral, 3, 73, 179, 192, 276 Sym formula, 98 tame link, 163, 169 tantrix, 142 thickness (of curve), 165 thin shell, 325 torque balance, 179 total curvature, 142 total variation, 140 tractrix, 101 triangulation, 11, 179, 255 condition for, 196 enumeration, see enumeration of triangulations irreducible, 235, 238 neighborly, see neighborly surface vertex-minimal, 237 triply orthogonal system, 19 turning angle, 142 Tutte’s embedding theorem, 318 twisted tractrix, 102 discrete, 107 vector area, 178, 181 vector bundle (discrete), 184 vertex split, 235 Voss surface (discrete), 77 wedge product, 309 Whitney form, 314 Wienholtz’s projection theorem, 154 Willmore energy, 183 discrete, writhe, 176, 185 zero-curvature condition, 86–89, 97–112, 124, 131 ... spheres (and circles), curvature-line parametrization, conformal parametrization, isothermic parametrization (conformal curvature-line parametrization), the Willmore functional (see Section 2) For... circular quadrilateral 18 Alexander I Bobenko There are deep reasons to treat circular nets as a discrete curvature-line parametriza- tion The class of circular nets as well as the class of curvature-line... surfaces (such as those with constant curvature), cubical complexes (including quad-meshes), discrete analogs of special parametrization of surfaces (such as conformal and curvature-line parametrizations),